When I do not want the link to Contact information I am writing the name in Photos slightly different from the name in the Contacts. For example, I am adding a prefix before the name, e.g. I am prefixing the names of my sailing friends with "Sailing: ", like "Sailing: Popeye".
Using names that are breaking the connection to your Contacts will prevent Photos from shortening the name to the first name only, so we will always be able to see the full name in Photos, and Photos will no longer be able to use the contact information to create smart location names like "Home of Grandpa", also some kinds of Memories will no longer be created, like birthday memories. Apple AI is using the Contacts to make Memories more personal, but the information is teated confidentially and will stay only local in our accounts. and I have never seen Photos trying to search more information about the named People & Pets on Facebook or across other social media, based on the Contact card.